Typical State-Space Diagonalization Procedure

As discussed in [452, p. 362] and exemplified in §C.17.6, to diagonalize a system, we must find the eigenvectors of $A$ by solving

$$A\underline{e}_i = \lambda_i \underline{e}_i$$

for $\underline{e}_i$ , $i=1,2$ , where $\lambda_i$ is simply the $i$ th pole (eigenvalue of $A$ ). The $N$ eigenvectors $\underline{e}_i$ are collected into a similarity transformation matrix:

$$E= \left[\begin{array}{cccc} \underline{e}_1 & \underline{e}_2 & \cdots & \underline{e}_N \end{array}\right]$$

If there are coupled repeated poles, the corresponding missing eigenvectors can be replaced by generalized eigenvectors.^2.12 The $E$ matrix is then used to diagonalize the system by means of a simple change of coordinates:

$$\underline{x}(n) \isdef E\, \tilde{\underline{x}}(n)$$

The new diagonalized system is then

$$\tilde{\underline{x}}(n+1) = \tilde{A}\, \tilde{\underline{x}}(n) + {\tilde B}\, \underline{u}(n)$$

$$\underline{y}(n) = {\tilde C}\, \tilde{\underline{x}}(n) + {\tilde D}\,\underline{u}(n),$$ (2.13)

where

$$\tilde{A} = E^{-1}A E$$

$${\tilde B} = E^{-1}B$$

$${\tilde C} = C E$$

$${\tilde D} = D. \protect$$ (2.14)

The transformed system describes the same system as in Eq.(1.8) relative to new state-variable coordinates $\tilde{\underline{x}}(n)$ . For example, it can be checked that the transfer-function matrix is unchanged.